用机器学习从马斯形式系数推断弗里克符号，准确率达96%。
Learning Fricke signs from Maass form Coefficients
- 通过平均同弗里克符号的傅里叶系数发现类似‘群体舞’的模式。
- 对已知符号的马斯形式分类准确率达96%（偶数）和94%（奇数）。
- 适用于数论研究者，尤其关注自守形式与数据驱动方法结合。
本文对马斯形式进行数据科学探究。发现具有相同弗里克符号的马斯形式的傅里叶系数经平均后呈现类似最近发现的‘群体舞’现象的模式，且引入奇偶性作为额外特征后该模式更加显著。我们数据集中约43%的形式弗里克符号未知。对剩余形式，采用线性判别分析（LDA）进行机器学习，对偶数奇偶性形式分类准确率为96%，奇数奇偶性为94%。将训练好的LDA模型应用于未知符号形式以预测符号。基于预测符号计算的平均值与已知符号形式进行对比，验证了预测的合理性。此外，部分预测结果与海哈尔算法提供的启发式猜测对比，约95%匹配。还使用神经网络获得与LDA相当的结果。
原文摘要 · Abstract (English)
In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered "murmuration" phenomenon, and that these patterns become more pronounced when parity is incorporated as an additional feature. Approximately 43% of the forms in our dataset have an unknown Fricke sign. For the remaining forms, we employ Linear Discriminant Analysis (LDA) to machine learn their Fricke sign, achieving 96% (resp. 94%) accuracy for forms with even (resp. odd) parity. We apply the trained LDA model to forms with unknown Fricke signs to make predictions. The average values based on the predicted Fricke signs are computed and compared to those for forms with known signs to verify the reasonableness of the predictions. Additionally, a subset of these predictions is evaluated against heuristic guesses provided by Hejhal's algorithm, showing a match approximately 95% of the time. We also use neural networks to obtain results comparable to those from the LDA model.
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